“My ex-husband and I agreed to end his child support payments permanently and I get to claim our child every year for taxes. Would both of us signing a notarized document of some sort be sufficient and not have legal repercussions on either one of us?”
In order for child support to be effectively discontinued, there must be an order of the court that either discontinues it or that says when in the future it ends. So, no, in most states a notarized document, on its own, would not be enough. You would need to have the notarized document entered into the court record, and accepted by the Court, with a short Court order that incorporates the notarized document and that orders the child support ended.